A new asymmetric extended family: Properties and estimation methods with actuarial applications

In the present work, a class of distributions, called new extended family of heavy-tailed distributions is introduced. The special sub-models of the introduced family provide unimodal, bimodal, symmetric, and asymmetric density shapes. A special sub-model of the new family, called the new extended heavy-tailed Weibull (NEHTW) distribution, is studied in more detail. The NEHTW parameters have been estimated via eight classical estimation procedures. The performance of these methods have been explored using detailed simulation results which have been ordered, using partial and overall ranks, to determine the best estimation method. Two important risk measures are derived for the NEHTW distribution. To prove the usefulness of the two actuarial measures in financial sciences, a simulation study is conducted. Finally, the flexibility and importance of the NEHTW model are illustrated empirically using two real-life insurance data sets. Based on our study, we observe that the NEHTW distribution may be a good candidate for modeling financial and actuarial sciences data.


Introduction
Probability distributions have been extensively used to model real-life data is several applied areas. For example, glass fiber, breaking stress of carbon fiber, and gauge lengths data [1] and wind speed data [2]. Speaking broadly, the heavy-tailed models play a vital role in analyzing and modelling data in several applied areas such as banking, risk management, financial and actuarial sciences, and economic, among others. The quality of statistical procedures mainly depends on the considered probability model of the studied phenomenon.
Usually the insurance and financial data sets are positive or right skewed [3,4], or unimodal shaped [5], and having heavy tails [6]. [7] showed that right-skewed data sets can be modeled using skewed distributions. Therefore, some unimodal and right-skewed distributions were a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 In the present work, an alternative approach to (1), we suggest a empirical approach to find heavy-tailed distributions. The proposed approach is based on the Monte Carlo simulation of the actuarial measures, namely value at risk (VaR) and tail value at risk (TVaR).
Let X represent the profit and loss distribution (profit positive and loss negative). The VaR at level q 2 (0, 1) is the lowest number, where the probability that Y = −X does not exceed y is at least 1 − q. Mathematically, VaR q (X) is the (1 − q) th quantile of Y, i.e., for detail see [21,22]. The TVaR is also referred to as tail conditional expectation (TCE). The TVaR determines the expected value of the losses under the condition that an event outside a certain probability level was occurred.
To apply the simulation approach based on (2) and (3), we first introduce a new family of distributions and then provide a simulation study of the VaR and TVaR for a special submodel of the proposed family to show empirically the heaviness of the tail of the proposed distribution.
Recently, [26] proposed a new wider family called extended Cordeiro and de Castro (ECC) family which is defined by the cdf G x; y; Z; p; ξ ð Þ ¼ 1 À 1 À Fðx; ξÞ y 1 À pFðx; ξÞ where y; Z > 0; ξ 2 R n ; and p 2 (0, 1). In the ECC family, the space of the parameter p is restricted to (0,1), hence the family with cdf (4) can not have enough flexibility to counter complex forms of data. Furthermore, the estimation of parameters of the special models of the ECC family as well as the computation of several distributional characteristics become very difficult due to the three additional parameters of the ECC family. Therefore, in this paper a flexible class of distributions, called the NEHT-F family is studied. The new family is proposed by keeping constant θ = 1 in (4) (to reduce the number of parameters), and instead of parameter p, we use a new parameter σ > 0 with unbounded upper bound unlike the upper limit of p.
The new extended class of heavy-tailed distributions is introduced via the T-X family approach of [27].
The T-X family approach is specified by the cdf where V[F(x;ξ)] satisfies three conditions as illustrated in [27].
The pdf corresponding to (6) is

Sub-models description
This section provides some sub-models from NEHT-F family called the NEHTW, NEHT-normal (NEHTN), and NEHTW-gamma (NEHTG) distributions. These sub-models can provide bimodal, unimodal, symmetric and asymmetric density shapes, and constant, increasing, modified bathtub, decreasing, J-shaped, and reversed-J shaped hazard rate functions.

NEHTW distribution
Consider the rv X that follows the one-parameter Weibull distribution with cdf Fðx; aÞ ¼ 1 À e À x a ; x � 0; a > 0, and pdf f ðx; aÞ ¼ ax aÀ 1 e À x a . Then, the cdf of the NEHTW distribution takes the form G x; Z; s; a ð Þ ¼ 1 À e À x a 1 À � sð1 À e À x a Þ � � Z ; x > 0; a; s; Z > 0: The corresponding pdf and hrf of the NEHTW model are g x; Z; s; a ð Þ ¼ asZx aÀ 1 e À Zx a f1 À � sð1 À e À x a Þg Zþ1 ; and h x; Z; s; a ð Þ ¼ asZx aÀ 1 f1 À � sð1 À e À x a Þg ; The QF of the NEHTW model is given by where 0 < p < 1. The one parameter Weibull distribution with parameter α follows as a special sub-model from the NEHTW distribution when η = σ = 1. For σ = 1, the two-parameter Weibull distribution follows as a special sub-model from the NEHTW model. The exponential distribution follows as a special case for α = σ = 1. For σ = 1 and α = 2, the NEHTW model reduces to the Rayleigh distribution.
For selected parameters values of the NEHTW distribution, some possible shapes for its density and hazard functions are sketched in Fig 1.

NEHTN distribution
The corresponding pdf and hrf of the NEHTN model are and h x; Z; s; a ð Þ ¼

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For selected parameters values of the NEHTN distribution, some possible shapes for its density and hazard functions are sketched in Fig 2, the shape of the NEHTN distribution can be unimodal, bimodal, right skewed or symmetry.

NEHTG distribution
Consider the rv X that follows the gamma (G) distribution with cdf Fðx; a; bÞ ¼ 1 where GðkÞ ¼ R 1 0 t aÀ 1 e À t dt is the G function and gðk; yÞ ¼ R y 0 t kÀ 1 e À t dt is the lower incomplete G function, and pdf f ðx; a; bÞ ¼ 1 Then, the cdf of the NEHTG distribution takes the form The corresponding pdf and hrf of the NEHTG model are For selected parameters values of the NEHTG distribution, some possible shapes for its density and hazard functions are sketched in Fig 3, the shape of the NEHTG distribution can be unimodal, bimodal, right skewed, symmetry or decreasing.

Two risk measures and numerical results
The actuaries are often interested in evaluating the exposure to market risks appear due to changes in some underlying variables including prices of equity, exchange rates or interest rates. This section is devoted to determining two important risk measures namely, VaR and TVaR of the proposed NEHTW distribution, which are useful in portfolio optimization.
The VaR is known in actuarial sciences as the quantile premium principle or quantile risk measure and it has been adopted by practitioners as a standard financial market risk. The VaR of a rv X is defined by the qth quantile of its cdf and it may be specified with a certain degree of confidence, q which is typically can be 90%, 95% or 99%, as shown in [28]. If X has cdf (8),

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then the VaR of X is given by where q is the qth quantile of X. The TVaR is an important measure to determine the expected value losses given that an event outside a certain probability level is occurred. Hence, the TVaR of the NEHTW model is specified by Inserting (9) in Eq (11), we can write Zsax aÀ 1 e À Zx a f1 À � sð1 À e À x a Þg Zþ1 dx: On solving, we get In the next section, based on (10) and (19), we provide a simulation study of VaR and TVaR to show empirically the heaviness of the tail of the NEHTW distribution.
In this section, numerical simulation results were presented for the VaR and TVaR of the two-parameter Weibull and the NEHTW distributions for several values of their parameters. The process was described below as follows: 1. Generating random samples of size n = 100 from both Weibull and NEHTW distributions.

Properties of the NEHT-G family
Some mathematical quantities of the NEHT-G family were introduced in this section.

Linear representation
Using the two power series and ð1 À zÞ The NEHT-G density can be expressed as gðx; Z; s; ξÞ ¼ Z s ; ξÞ jþk f ðx; ξÞ: where h (j+k+1) (x) is the exponentiated-G (exp-G) pdf with positive power parameter (j + k + 1) and

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Hence, the density function of the NEHT-G family can be expressed as a linear combination of exp-G densities. So, Eq (23) reveals that some mathematical quantities of the proposed family can be derived from the quantities of the exp-G family. Hereafter, let Y j+k+1 denote a rv having the exp-G density with power parameter (j + k + 1), then some properties of the rv X are expressed simply from those of Y j+k+1 .

Moments and generating function
The nth moment of the NEHT-G family was expressed from (23) as The rth incomplete moment of the NEHT-G family follows simply from (23) as The first incomplete moment (FIM) of X follows from (23) as R GðtÞ 0 u jþk Q G ðuÞdu is the FIM of the exp-G family which can be determined numerically or analytically from a baseline qf Q G (u) = G −1 (u;ξ).
The moment generating function (mgf) of the NEHT-G family was derived from (23) as where M j+k+1 (t) is the mgf of Y j+k+1 and pðt; j þ kÞ ¼ R 1 0 u kþj exp½t Q G ðuÞ�du. Then, M(t) follows from the exp-G mgf.

Rényi entropy
The Rényi entropy (RE) of a rv X is a measure of variation of uncertainty. The RE is given by ; u > 0 and u 6 ¼ 1: Then, the RE of the NEHT-G family takes the form
Hence, the pdf of the sth order statistic for the NEHT-G family is a linear combination of exp-G densities as follows g s:n ðxÞ ¼ where h j+k+1 (x) is the pdf of the exp-G class with power parameter (j + k + 1) and Using (25), some mathematical quantities of X s:n follows simply from the properties of Y j+k+1 .

Estimations
To estimate the parameters of the NEHT-F family, this section assigns the maximum likelihood estimators (MLEs) to estimating and to provide some simulations to explore the behavior of the MLEs.

Maximum likelihood estimation
Suppose that x 1 , x 2 , . . ., x n are given values of a random sample of size n from the NEHT-F family with parameters (η, σ, ξ). The log-likelihood function takes the form The partial derivatives of (26) are By setting @' @Z ¼ 0, @' @s ¼ 0, and @' @ξ ¼ 0; one can solve them numerically to obtain the MLEs of η, σ and ξ, respectively.
These nonlinear system of equations can be solved using any statistical software.
Suppose that x 1 , x 2 , . . ., x n are given values of a random sample of size n from the NEHTW distribution with parameters α, σ and η.

Maximum likelihood
The log-likelihood function for the NEHTW model in (9) is given by where Θ = (α, σ, η) > . The function provided in (30) can be numerically solved by using Newton-Raphson method (iteration method). The partial derivatives of (30), with respect to the parameters, are

Ordinary and weighted least-squares
The OLSE of the NEHTW parameters can be obtained by minimizing the following function with respect to α, σ and η, Further, the OLSE of the NEHTW parameters can be obtained by solving the non-linear equation where The WLSE of the NEHTW parameters are obtained by minimizing the following with respect to α, σ and η. Also, the WLSE can be obtained by solving the non-linear equation where Δ 1 (�|α, σ, η), Δ 2 (�|α, σ, η) and Δ 3 (�|α, σ, η) are defined in Eq (31).

Percentile
From (11), the PCE of the parameters of NEHTW model can be obtained by minimizing the following function with respect to α, σ and η.

Simulation study
To assessing the performance of the eight estimates aforementioned for the NEHTW parameters, we devote this section. The simulation results is carried out for the NEHTW distribution by the function optim () with the argument method = "L-BFGS-B" available by R software (version 4.0.5) [29]. In addition, Tables 3-7 show the rank of each estimator among all the remaining estimators in each row, with partial sum of the ranks for each column (∑ Ranks) for each sample size.
The partial and overall ranks of all estimators are shown in Table 8. From Table 8, we can conclude that the MPSE outperforms all the other estimators with an overall score of 142. Thus, for the NEHTW distribution we recommend the MPSE estimation method.

Applications and numerical computations of VaR and TVaR
The key application of the heavy-tailed distributions is the insurance loss phenomena or extreme value theory. Here, we consider two data sets from insurance sciences. The flexibility of the NEHTW model was illustrated by analyzing two insurance data sets. Furthermore, corresponding to the analysed data sets, we calculate the VaR and TVaR of the NEHTW and      Weibull models to show empirically, using the two real data sets, that NEHTW model has a heavier tail than, its sub-model, Weibull distribution. The comparison of the NEHTW distribution is made with some important distributions including two-parameter Weibull, Lomax, Burr-XII (BX-II), exponentiated Lomax (EL) and beta Weibull (BW) distributions.
To decide about the best fitting among the competitive distributions, certain comparative tools called, AIC, BIC, Cramer Von Mises (CM), Anderson Darling (AD), and KS statistic with its corresponding p-value are taken in to account. The formulae of these discrimination measures can be explored in [30].

Applications to earthquake insurance and vehicle insurance losses data
In this subsection, we consider two practical insurance applications to show the flexibility of the NEHTW model in practice. 9.1.1 The earthquake insurance data. We begin with our first illustration by considering a heavy-tailed data set which refers to the earthquake insurances. This data set is available online at: https://earthquake.usgs.gov/earthquakes. The MLEs with standard errors (in parentheses) of the parameters and the estimated values of the considered measures were reported in Tables 9 and 10. Based on the values in Table 10, we conclude that the NEHTW model

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provide adequate fits as compared to other competitors. So, we prove empirically that NEHTW distribution can be a better model than other competitive models. Furthermore, corresponding to the earth quick insurance data, the estimated pdf and cdf of the NEHTW model were displayed in Fig 5. The Kaplan Meier survival (KMS) and PP plots are presented in Fig 6. The plots show that the NEHTW model has the best fitting to the earthquake insurance data. 9.1.2 The vehicle insurance loss data. The second data set is about vehicle insurance losses and it is available online at the following URL: https://data.world/datasets/insurance. The parameters estimates and associated standard error (in parentheses) of the NEHTW model and other models were listed in Table 11. The values of analytical measures for the NEHTW and other studied models were reported in Table 12. The values in Table 12 show

The VaR and TVaR based on the two insurance data sets
This subsection is devoted to exploring the VaR and TVaR measures of the NEHTW and Weibull distributions via the estimated parameters of the two analyzed insurance data sets previously discussed in Section 7.1. The empirical results were listed in Tables 13 and 14

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The results for the two actuarial measures, Var and TVaR, of the NEHTW and Weibull distributions, which were reported in Tables 13 and 14, show that the NEHTW distribution has a heavier tail than the tail of Weibull distribution and hance it can be used for modelling heavytailed insurance data.

Conclusions
In this article, a new family of heavy-tailed distributions is proposed and studied. A special sub-model of the introduced family called, the new extended heavy-tailed Weibull (NEHTW) distribution is discussed in detail. The introduced NEHTW model is very flexible and can be adopted effectively to model data with heavy tails which encountered in insurance science.

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The maximum likelihood and other seven estimation approaches are adopted to estimate the NEHTW parameters. Their performances are assessed using detailed simulation results which are ranked to explore the best estimation approach for the model parameters. Based on our study, the maximum product of spacing is recommended to estimate the NEHTW parameters. Two important risk measures of the NEHTW model are studied numerically and empirically using a real-life insurance data. Both numerical and empirical studies of the two risk measures show that the introduced NEHTW distribution has heavier tail than its sub-model and can be used quite effectively in modelling heavy tailed insurance data sets. Finally, practical applications to insurance data sets are analyzed and the comparisons of the NEHTW model are made with some other important competitors. The practical applications and simulation of the two actuarial measures show that the NEHTW distribution is a good alternative for modelling heavy-tailed insurance data.
There are some possible future extensions to this study, such as a discrete version of the proposed family can be introduced following the work of [31]. Furthermore, a survival regression extended heavy-tailed Weibull model can be addressed for complete and censored data.